Accelerometers can find applications in many areas of technologies. For example, in the automotive industry, acceleration sensing is commonly used for airbag deployment. The computer industry utilizes accelerometers to protect hard disks from large shocks, and the aerospace industry employs inertial measurement units comprising multiple accelerometers and gyroscopes for sensing and navigation. Accelerometers are also used in many personal handheld devices as well, where they can detect the general orientation of the devices.
In many high volume applications, accelerometer devices are made using microelectromechanical system (MEMS) fabrication technologies. These techniques allow batch fabrication in a CMOS process flow and can have the benefit of reductions in size, weight, power, and cost (SWaP-C) while maintaining adequate performance for a variety of applications.
Conventional MEMS accelerometers usually measure the electric charge on a capacitor to detect small movements of a proof mass attached to a spring so as to derive the acceleration of the proof mass. However, it can be challenging for a conventional MEMS accelerator to detect acceleration on the order of sub milliG (1 G=9.8 m/s2) because this level of acceleration may only generate nanovolt changes that are difficult to measure with high precision.
Optically-enabled accelerometers, where the capacitive pickoffs are replaced with an optical transducer, can address the limits of capacitive accelerometers. Existing optical approaches typically rely on measuring small displacements of a mechanical proof mass and translating these displacements into acceleration. Therefore, the sensitivity of an optical accelerator accelerometer depends on the precision of the optical measurement system.
Displacement-based accelerometers can have resolutions down to 10−9 g, but they also suffer several limitations. First, any small displacement arising from thermal expansion, packaging stress, acceleration in an orthogonal dimension, or other unwanted drift can also be picked up by the measurement system and erroneously translated into acceleration readings. Second, some optically-enabled accelerometers exploit evanescent optical coupling to measure minute displacements, which can place restrictions on the scale factor stability and full scale linear dynamic range of the device when operating in open-loop mode. Third, optical techniques that use highly sensitive interferometric measurement typically also use optical sources with high levels of wavelength stability and precision, creating a significant challenge to be applied in small form factors and in harsh environments.
FIG. 1A shows a schematic of a displacement-based accelerometer 101 including a proof mass 111 suspended by a pair of tethers 121a and 121b from two anchors 141a and 141b. Each side of the proof mass 101 also has a respective measuring element 131a and 131b, connected to the anchors 141a and 141b, to measure the displacement of the proof mass 101 with respect to the anchors 141a and 141b. The accelerometer 101 is limited by the described above trade-off among sensitivity, stability, and dynamic range.
Resonant accelerometers (also referred to as frequency-modulated accelerometers) can relieve the constraints in displacement-based accelerometers by sensing acceleration based on detection of the resonant frequency of the tethers that suspend the proof mass. Acceleration of the proof mass causes opposing changes in the effective stiffness of the tethers, resulting in equal but opposite shifts in their resonant frequencies. Detection of this opposing shift can be utilized to calculate the acceleration of the proof mass, while any mutual shift of the tethers caused by unwanted orthogonal acceleration or temperature drift are cancelled out.
FIG. 1B shows a schematic of a resonant accelerometer 102 including a proof mass 112 suspended from two anchors 142a and 142b by two tethers 122a and 122b, respectively. A pair of vibrating sensing tethers 132a and 132b, including electrostatic comb drives, are also attached between the proof mass 112 and the anchors 142a and 142b. 
In operation, the proof mass 102 experiences displacement as a result of applied acceleration. The displacement of the proof mass 102 pulls one of the tethers (e.g., 132a) into tension while pushing the other tether (e.g., 132b) into compression, thereby altering the resonant frequencies of the tethers 132a and 132b. The resonant frequency shifts have equal magnitudes but opposite signs when the acceleration of the proof mass 102 occurs along the desired axis. Any acceleration, and resulting displacement, experienced in orthogonal dimensions can force the tether resonant frequencies to shift together, which allows for a differential measurement and a cancellation of unwanted signals.
The two vibrating sensing tethers 132a and 132b can be excited and detected using the electro-static comb drives. These comb drives can be used to both excite motion in the vibrating sensing tethers 132a and 132b, typically at their natural mechanical resonant frequency, as well as to detect this induced motion including measurement of the resonant frequencies of the vibrating sensing tethers 132a and 132b. 
The sensitivity of the resonant accelerometer 102 can be described by the scale factor, or the amount of frequency shift experienced by an individual sensing tether 132a/132b as a result of a given acceleration of the proof mass 102 (e.g., in units of Hz/g). Generally, a larger scale factor is desirable, not only to increase system sensitivity, but also to reduce the impact of unwanted drift in the sensor signal due to temperature and other fluctuations in the surrounding environment over time. For example, in the case where the scale factor is equal to 10 Hz/g and the tether resonant frequency is stable to within 1 Hz over long periods of time, the measured signal, in units of measured acceleration, usually drifts by 0.1 g over this time. If instead the scale factor is increased to 10 kHz/g and the tether frequency stability stays exactly the same, the measured signal may drift by only 0.1 mg.
The scale factor depends on the ratio of the size of the proof mass 102 to the size of the tethers 132a/132b, where larger proof masses and smaller tethers can result in larger scale factors. The size of the sensing tethers 132a/132b is typically limited by the electro-static comb drives that both excite and detect their motion. Smaller tethers can suffer from reduced detection sensitivity, which is dependent on the surface area of the comb. This reduced sensitivity, combined with smaller displacement amplitudes, can make it very difficult to monitor motions of tethers with cross-sectional dimensions of less than 10 microns. This limits the achievable scale factor in conventional MEMS based resonant accelerometers.